The Principle of Minimum Justified Correlation
Abstract
It is shown that Shannon entropy, Fisher information, and quantum-mechanical kinetic energy may all be viewed as measures of correlation. We begin with a fundamental correlation-destroying map: a joint probability density ρ(x,y) is replaced by ρx(x)ρy(y), the product of its marginal distributions. In the discrete case, Shannon's entropy is nondecreasing under this map. For continuous distributions, this fundamental map leads to a well-behaved, coordinate-invariant correlation measure, analogous to the discrete Shannon entropy: I[ρ] = h[ρx] + h[ρy] − h[ρ]. In the continuous case, however, another measure appears: Fisher information. The relative Fisher information J(ρ|ρxρy) behaves similarly under the fundamental map. For a normalized real quantum wavefunction, this decrease is exactly proportional to the decrease in mean kinetic energy, ⟨ T⟩ψ− ⟨ T⟩Φ= ℏ2/8m J(ρ|ρxρy), Φ= √(ρx ρy). In Jaynes's language, the result supports a principle of minimum justified correlation: given physical constraints and a set of possible distributions or related amplitudes satisfying those constraints, select from that set those with the least correlation. This is Part I of a two-part paper. Here we develop the entropy, Fisher-information, and kinetic-energy identities above, and state the principle they support. Part II addresses questions which Part I raises but leaves unanswered: multiple solutions, time dependence, the role of spin, a route to the Schrödinger equation itself, and a proposed experimental test. AI has been used.
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BibTeX
@misc{drie2026principle,
title = {{The Principle of Minimum Justified Correlation}},
author = {John H Van Drie},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.38124},
eprint = {2609.38124},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.38124},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The Principle of Minimum Justified Correlation
AU - John H Van Drie
PY - 2026
DA - 2026/09/29
PB - arXiv
DO - 10.48550/arxiv.2609.38124
UR - https://arxiv.org/abs/2609.38124
AB - It is shown that Shannon entropy, Fisher information, and quantum-mechanical kinetic energy may all be viewed as measures of correlation. We begin with a fundamental correlation-destroying map: a joint probability density ρ(x,y) is replaced by ρ_x(x)ρ_y(y), the product of its marginal distributions. In the discrete case, Shannon's entropy is nondecreasing under this map. For continuous distributions, this fundamental map leads to a well-behaved, coordinate-invariant correlation measure, analogous to the discrete Shannon entropy: I[ρ] = h[ρ_x] + h[ρ_y] − h[ρ]. In the continuous case, however, another measure appears: Fisher information. The relative Fisher information J(ρ|ρ_xρ_y) behaves similarly under the fundamental map. For a normalized real quantum wavefunction, this decrease is exactly proportional to the decrease in mean kinetic energy, ⟨ T⟩_ψ− ⟨ T⟩_Φ= ℏ^2/8m J(ρ|ρ_xρ_y), Φ= √(ρ_x ρ_y). In Jaynes's language, the result supports a principle of minimum justified correlation: given physical constraints and a set of possible distributions or related amplitudes satisfying those constraints, select from that set those with the least correlation. This is Part I of a two-part paper. Here we develop the entropy, Fisher-information, and kinetic-energy identities above, and state the principle they support. Part II addresses questions which Part I raises but leaves unanswered: multiple solutions, time dependence, the role of spin, a route to the Schrödinger equation itself, and a proposed experimental test. AI has been used.
ER -